How to Write a Proper Fraction in Lowest Terms

Proper Fraction

A fraction is called a proper fraction if its numerator is less than its denominator. The value of a proper fraction is always less than 1. For example, Sam got a chocolate bar and he divided it into four equal parts. He took one part and gave three parts to his sister Rachel. We represent Sam's portion as 1/4 and Rachel's portion as 3/4. Both these fractions are known as proper fractions because here the numerator is less than the denominator.

1.	What is Proper Fraction?
2.	Adding Proper Fractions
3.	Subtracting Proper Fractions
4.	Multiplying Proper Fractions
5.	Dividing Proper Fractions
6.	FAQs on Proper Fraction

What is Proper Fraction?

A fraction in which the numerator value is always lesser than the denominator value is known as a proper fraction. For example, 18/25, 19/45, 62/78, 1/6, 1/9, are proper fractions. A fraction consists of two parts, the numerator and the denominator and the various kinds of fractions are identified on the basis of these two values. The two primary fractions which are distinguished on this criterion are proper and improper fractions.

$Proper fraction with a smaller numerator and a larger denominator$

Difference Between Proper, Improper, and Mixed fractions

Proper fraction
A fraction in which the numerator is less than the denominator is termed as a proper fraction. In other words, when the result of a fraction is less than 1, then that fraction is a proper fraction. For example, 5/8 and 4/7 are proper fractions because 5 < 8 and 4 < 7.

Improper fraction
A fraction in which the numerator is greater than or equal to the denominator is known as an improper fraction. In this, the result of a fraction is greater than or equal to 1. For example, 7/2 and 6/6 are improper fractions because 7 > 2 and 6 = 6.

Mixed fraction
A mixed fraction is a combination of a whole number and a proper fraction. For example, in $3\dfrac{2}{9}$, 3 is a whole number and 2/9 is a proper fraction.

Adding Proper Fractions

In order to add two proper fractions, we add the numerators if the given fractions are like fractions, i.e., if the given denominators are the same. For example, 2/8 + 3/8 can be easily added by just adding the numerators since the denominators are the same. The sum of 2/8 + 3/8 will be 5/8. However, to add proper fractions that are unlike, in which the denominators are different, we take the LCM (Least Common Multiple) of the denominators and rewrite the fractions as equivalent fractions using the LCM as the common denominator. Now, when all the denominators become the same, we add the numerators and write the result as the final numerator on top of the common denominator. For example, to add 2/3 + 4/5, we take the LCM of the denominators. The LCM of 3 and 5 is 15. Now, we multiply both the fractions with such a number so that the denominators remain the same. This results in (10 + 12)/15 = 22/15.

Subtracting Proper Fractions

The subtraction of proper fractions is similar to addition. If we need to find the difference of proper fractions that are like fractions, we simply find the difference of the numerators, keeping the same denominator. For example, the difference of 6/9 - 4/9 will be 2/9.

Now, if we want to subtract proper fractions that are unlike, in which the denominators are different, we take the LCM (Least Common Multiple) of the denominators and rewrite the fractions as equivalent fractions using the LCM as the common denominator. When all the denominators become the same, we subtract the numerators and write the result on the common denominator.

For example, to subtract 8/9 - 3/4, we take the LCM of the denominators. The LCM of 9 and 4 is 36. Now, we multiply both the fractions with such a number so that the denominators are the same. This results in (32 - 27)/36 = 5/36.

Multiplying Proper Fractions

Unlike addition and subtraction, the multiplication and division of proper fractions is easier in a way. We simply multiply the given numerators, then multiply the denominators, and finally simplify or reduce the resultant fraction. For example, to multiply 2/6 × 5/4, we multiply the numerators 2 and 5, to get 10, and we multiply the denominators 6 and 4 to get 24. The product is written as 10/24 which can be further reduced to 5/12.

Dividing Proper Fractions

The division of proper fractions is similar to multiplication. The only difference is that we change the division sign to the multiplication sign and then we multiply the first fraction with the inverse of the second fraction. For example, let us divide: 4/9 ÷ 2/3. This will become 4/9 × 3/2 =12/18. This can be further reduced to 2/3.

Converting Improper Fractions to Proper Fractions

Mathematically, an improper fraction can be converted to a mixed fraction, which is the combination of a whole number and a proper fraction. For example, let us convert an improper fraction 13/5 to a mixed fraction by using the following steps.

Divide the numerator by the denominator. Here, on dividing 13 ÷ 5, we get 2 as the quotient and 3 as the remainder.
Write the obtained quotient as the whole number and the remainder as the numerator on the same denominator. Here, the quotient (2) will be the whole number, the remainder (3) will be the new numerator and the denominator will be the same.
This converts the given improper fraction to a mixed fraction which has a whole number and a proper fraction. Therefore, 13/5 will be written as $ 2\dfrac{3}{5}$

How to Add Mixed Fraction to Proper Fraction?

To add a mixed fraction to a proper fraction we simply convert the mixed fraction into an improper fraction and then we add the two fractions in the usual way of the addition of fractions. After finding the answers we again convert the result into a mixed fraction.

For example, let us add $ 2\dfrac{1}{5}$ + $\dfrac{1}{4}$

Convert the mixed fraction into an improper fraction that is $ 2\dfrac{1}{5}$ = 11/5

11/5 + 1/4 = (44 + 5)/20 = 49/20

Now, convert 49/20 into a mixed fraction which will be = $ 2\dfrac{9}{20}$

Check out the interesting topics related to proper fractions.

Fractions
Improper Fraction
Types of Fractions

Proper Fraction Examples

Example 1: Identify the proper fractions from the following:

a.) 17/25

b.) 18/13

c.) 16/16

Solution:

a.) 17/25 is a proper fraction because the numerator is less than the denominator.

b.) 18/13 is not a proper fraction because the numerator is greater than the denominator. It is an improper fraction.

c.)16/16 is not a proper fraction because the numerator is equal to the denominator. It is an improper fraction.
Example 2: Add the proper fractions: 5/13 + 4/13.

Solution:

To add the given fractions, we add the numerators and keep the denominators the same. 5 + 4 = 9. Therefore, the sum of the fractions is 9/13.
Example 3: Convert the given improper fraction to a proper fraction: 14/5

Solution:

An improper fraction can be converted to a mixed fraction which is the combination of a whole number and a proper fraction. To convert 14/5 to a mixed fraction, we will divide 14 ÷ 5. This will give 2 as the quotient and 4 as the remainder. So, the quotient (2) will become the whole number, and the remainder (4) will become the numerator. This will be written as a mixed fraction $ 2\dfrac{4}{5}$.
Example 4: Multiply the proper fractions: 4/5 × 2/6.

Solution:

To multiply 4/5 × 2/6, we will first multiply the given numerators, then multiply the denominators, and finally, reduce the fraction. For example, to multiply 4/5 × 2/6, we multiply the numerators 4 and 2, to get 8, and we multiply the denominators 5 and 6 to get 30. The product is 8/30 which can be further reduced to 4/15.

go to slidego to slidego to slidego to slide

Breakdown tough concepts through simple visuals.

Math will no longer be a tough subject, especially when you understand the concepts through visualizations.

Book a Free Trial Class

Practice Questions on Proper Fraction

go to slidego to slide

FAQs on Proper Fraction

What is a Proper Fraction in Math?

The fraction in which the numerator is less than the denominator is called a proper fraction. For example, 7/12, 7/8 are proper fractions.

What is the Difference Between Proper Fraction and Improper Fraction?

The fraction in which the denominator is greater than the numerator is called a proper fraction. For example, 3/4, 27/39 are proper fractions. On the other hand, a fraction in which the numerator is greater than or equal to the denominator is called an improper fraction. For example, 18/5, 49/23, 46/46 are improper fractions.

How to Convert Improper Fraction to Proper Fraction?

Mathematically, an improper fraction can be changed to a mixed fraction which is the combination of a whole number and a proper fraction. For example, let us convert an improper fraction 17/2 to a mixed fraction by using the following steps.

Divide the numerator by the denominator. Here, on dividing 17 by 2, we get 8 as the quotient and 1 as the remainder.
Write the obtained quotient as the whole number and the remainder as the numerator on the same denominator. Here, the quotient (8) will become the whole number, 1 will be the new numerator and the denominator will be the same.
Now, the given improper fraction is converted to a mixed fraction which has a whole number and a proper fraction. Therefore, 17/2 will be written as $8\dfrac{1}{2}$

How to Add Proper Fractions with Different Denominators?

To add proper fractions with different denominators, we take the LCM (Least Common Multiple) of the denominators and rewrite the fractions as equivalent fractions using the LCM as the common denominator. Now, when all the denominators become the same, we add the numerators and write the sum as the final numerator on the common denominator. For example, to add 1/2 + 2/3, we take the LCM of the denominators. The LCM of 2 and 3 is 6. Now, we multiply both the fractions with such a number so that the denominators are the same. This results in (3 + 4)/6 = 7/6. This improper fraction can be finally converted to a mixed fraction $1\dfrac{1}{6}$.

How to Multiply Proper Fractions?

In order to multiply proper fractions, we multiply the given numerators, then multiply the denominators, and then reduce the fraction to its lowest terms. For example, to multiply 4/9 × 3/6, we multiply the numerators 4 and 3, to get 12, and we multiply the denominators 9 and 6 to get 54. The product is 12/54 which can be further reduced to 2/9.

How to Divide Proper Fractions?

We divide proper fractions in the same way as we multiply them. The only difference is that we multiply the first fraction with the reciprocal (inverse) of the second fraction. For example, let us divide: 2/3 ÷ 4/5. We will write the reciprocal of the second fraction and then multiply the fractions. This will give us 2/3 × 5/4 = 10/12. This can be further reduced to 5/6.

Is 12/8 a Proper Fraction?

No, 12/8 is not a proper fraction because the numerator (12) is greater than the denominator (8). It is an improper fraction.

Is 7/7 a Proper Fraction?

No, 7/7 is not a proper fraction because the numerator is equal to the denominator. It is considered to be an improper fraction.

Is 5/8 a Proper Fraction?

Yes, 5/8 is a proper fraction because the numerator (5) is less than the denominator (8). Thus, 5/8 is a proper fraction.

How to Write a Proper Fraction in Lowest Terms

Source: https://www.cuemath.com/numbers/proper-fraction/